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Unformatted text preview: ideal
characteristics (Fig. 7.16). J( 2
_ w + 142122.3
2 + 142122.3)2 + (753.98w cos 0)2
w The closer the poles are to the zeros (closer the 0 t o ~), the faster the gain recovery from
1 on either side of Wo = 1207r. Figure 7.17b shows the amplitude response for three
different values of O. This example is a case of very simple design. To achieve zero gain
over a band, we need an infinite number of poles as well as of zeros. • o to o C omputer E xample C 7.3
Plot the amplitude response of the transfer function
H (s) = 2 of a second order notch filter for Wo = 1207r and 7 .44 Notch (Bandstop) Filters An ideal n otch filter amplitude response (shown shaded in Fig. 7.17b) is a
complement of t he a mplitude response of a n ideal bandpass filter. I ts gain is zero
over a small b and centered a t some frequency Wo a nd is unity over t he remaining
frequencies. Realization of such a characteristic requires a n infinite number of poles
and zeros. Let us consider a practical secondorder notch filter to obtain zero gain a t
a frequency w = woo For this purpose we m ust have zeros a t ± jwo. T he r equirement
of u nity gain a t w = 0 0 requires t he n umber of poles to be equal to t he n umber
of zeros (m = n ). T his ensures t hat for very large values of w , t he p roduct of t he
d istances of poles from w will be equal t o t he p roduct of the distances of zeros from
w . Moreover, u nity g ain a t w = 0 requires a pole a nd t he corresponding zero t o
be equidistant from t he origin. For example, if we use two (complex conjugate)
zeros, we m ust have two poles; t he d istance from t he origin of t he poles a nd of
the zeros should b e t he s ame. T his requirement can be met by placing t he two
conjugate poles o n t he semicircle of radius wo, as depicted in Fig. 7.17a. T he poles
can b e a nywhere on t he semicircle t o satisfy t he equidistance condition. Let t he
two conjugate poles be a t angles ±O w ith respect to t he negative real axis. Recall
t hat a pole a nd a zero in t he vicinity tend t o cancel o ut each o ther's influences.
Therefore, placing poles closer t o zeros (selecting 8 closer to 7r / 2) r esults in a rapid
recovery of t he g ain from value 0 t o 1 as we move away from Wo in either direction.
Figure 7.17b shows t he gain I H(jw)1 for three different values of 8.
• E xample 7.5
Design a secondorder notch filter to suppress 60 Hz hum in a radio receiver. 2 s + wo
s2 + (2wo cos 8 )s + w5
0 = 60°,80°, and 87°. w O=120*pi;
t heta=[60 80 8 7j*(pi/180);
for m =I:length(theta)
n um=[1 0 w O'2];
d en=[1 2 *wO*cos(theta(m» wO'2j;
w=0:.5:1000; w =w';
[ mag,phase,wj=bode(num,den,w);
p lot(w,mag),hold on,axis([O 1000 0 1.1])
e nd
0
Figures 7.16b a nd 7.17b show t hat a n otch (stopband) filter frequency response
is a complement of t he b andpass filter frequency response. I f H B P( s ) a nd H B S( s)
are t he t ransfer functions of a bandpass a nd a b andstop filter (both centered a t t he
same frequency), then
H BS(S) = 1  HBP(S) Therefore, a ba...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.
 Spring '13
 Bayliss
 Signal Processing, The Land

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